{
  "abstract": "In [Combinatorica 1996], Reed, Robertson, Seymour and Thomas proved Younger's conjecture dating back to 1973 that states that directed cycles have the Erdős-Pósa property. Specifically, they showed that every directed graph $D$ contains either $k$ arc-disjoint cycles or a set of at most $f(k)$ arcs that intersect every directed cycle. The function $f$, depending only on $k$, is however notoriously large--in fact, not even elementary. The first improvement on this function was only very recently provided by Hatzel, Kreutzer, Milani, and Muzi [FOCS 2024] who showed that $f$ can be taken to be an exponential tower of height $8$. In this work, we improve this function to $O(k^4)$ in the case where the digraph is Eulerian. This also answers an open question by Wahlström [Springer Monographs in Mathematics 2018] who asked explicitly when Eulerian digraphs have a polynomial Erdős-Pósa property. The only other known result about Erdős-Pósa on Eulerian digraphs is a result from Seymour [Combinatorica 1996] who showed that $f(k)=k$ when the Eulerian digraph admits a linkless embedding in 3-space, and such an equality is not possible for general Eulerian digraphs. We also complement our positive result, by showing that the result of Seymour cannot be extended to give a linear bound ($O(k)$) on general Eulerian digraphs, by showing that even on Eulerian digraphs the lower bound for the Erdős-Pósa property is $Ω(k \\log k)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06250",
  "author_total": 3,
  "authors": [
    {
      "name": "Esther Galby"
    },
    {
      "name": "Roohani Sharma"
    },
    {
      "name": "Bart van der Steenhoven"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a5m4xkg7783zd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06250",
  "fulltext_url": null,
  "id": "a5m4xkg7783zd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06250",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06250",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a5m4xkg7783zd.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06250",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06250"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Polynomial Erdős-Pósa bound for Eulerian digraphs",
  "title_text": "A Polynomial Erdős-Pósa bound for Eulerian digraphs",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/a5m4xkg7783zd",
  "venue": "arXiv",
  "versions": []
}